Causally Stable Approximation of Optimal Maps in Maximal Value Constrained Least-Squares Optimization - Tanovic, Omer; Megretski, Alexandre (2019 18th European Control Conference (ECC), 2019)
In this paper, we consider a problem of designing discrete-time systems which are optimal in frequency-weighted least squares sense subject to a maximal output amplitude constraint. In such problems, the optimality conditions do not provide an explicit way of generating the optimal output as a real-time implementable transformation of the input, due to causal instability of the resulting dynamical equations and sequential nature in which criterion function is revealed over time. On the other hand, under some mild conditions, the optimal system has exponentially fading memory which suggests existence of arbitrarily good finite-latency approximations. In this paper, we extend the method of balanced truncation for linear systems to the class of nonlinear models with weakly contractive operators. We then propose a causally stable finite-latency nonlinear system which returns high-quality approximations to the optimal map. The proposed system is obtained by a careful truncation of an infinite dimensional state space representation of the optimal system, as suggested by the derived generalization of the balanced truncation algorithm.